Exact inverses of certain band matrices

Allgower, Eugene L.

doi:10.1007/bf01436382

Cited by 21 publications

(9 citation statements)

References 3 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is possible to find an inverse to the covariance matrix, eq(24), [39], and by the spectral theorem a real symmetric matrix is diagonalizable by orthogonal matrices. Thus we can argue, as in [26], that there is a orthogonal matrix P such that P Σ P − 1 = Λ where is the covariance matrix, eq(24), where Δ j is the column vector with displacement of distance j and Λ is a diagonal matrix with diagonal elements described by the vector λ .…”

Section: D Examplesmentioning

confidence: 99%

Real-time single-molecule 3D tracking in E. coli based on cross-entropy minimization

Amselem

Broadwater

Havermark

et al. 2022

Preprint

View full text Add to dashboard Cite

Sub-ms 3D tracking of individual molecules in living cells is an important goal for microscopy since it will enable measurements at the scale of diffusion limited macromolecular interactions. Here, we present a 3D tracking principle based on the true excitation point spread function and cross-entropy minimization for position localization of moving fluorescent reporters that approaches the relevant regime. When tested on beads moved on a stage, we reached 67nm lateral and 109nm axial precision with a time resolution of 0.84 ms at a photon count rate of 60kHz, coming close to the theoretical and simulated predictions. A critical step in the implementation was a new method for microsecond 3D PSF positioning that combines 3D holographic beam shaping and electro-optical deflection. For the analysis of tracking data, a new point estimator for diffusion was derived and evaluated by a detailed simulation of the 3D tracking principle applied to a fictive reaction-diffusion process in an E. coli-like geometry. Finally, we successfully applied these methods to track the Trigger Factor protein in living bacterial cells. Overall our results show that it is possible to reach sub-millisecond live-cell single molecule tracking, but that it is still hard to resolve state transitions based on diffusivity at this time scale.

show abstract

Section: D Examplesmentioning

confidence: 99%

Real-time single-molecule 3D tracking in E. coli based on cross-entropy minimization

Amselem

Broadwater

Havermark

et al. 2022

Preprint

View full text Add to dashboard Cite

show abstract

“…[Trench, 1974] Trench presents a method for inverting {p, q}-banded Toeplitz matrices by exploiting the banded structure. [Bevilacqua and Capovani, 1976] Bevilacqua and Capovani extend the results of the papers [Greenberg and Sarhan, 1959] and [Allgower, 1970[Allgower, , 1973 to band matrices and to block band matrices (not necessarily symmetric). Formulas are presented for inverting band matrices whose elements on the extreme diagonals are different from zero.…”

Section: The Overviewmentioning

confidence: 73%

“…[ Uppuluri and Carpenter, 1969] Uppuluri and Carpenter present an exact formula to compute the inverse of a specific covariance matrix, which is a symmetric tridiagonal Toeplitz matrix. [Allgower, 1973] Allgower provides a method for calculating the inverse of banded Toeplitz matrices. [Kershaw, 1970] Kershaw provides bounds between which the elements of the inverse of a tridiagonal matrix with positive off-diagonal elements will lie.…”

Section: The Overviewmentioning

confidence: 99%

A bibliography on semiseparable matrices*

Vandebril

Barel

Golub

et al. 2005

Calcolo

View full text Add to dashboard Cite

Currently there is a growing interest in semiseparable matrices and generalized semiseparable matrices. To gain an appreciation of the historical evolution of this concept, we present in this paper an extensive list of publications related to the field of semiseparable matrices. It is interesting to see that semiseparable matrices were investigated in different fields, e.g., integral equations, statistics, vibrational analysis, independently of-each other. Also notable is the fact that leading statisticians at that time Used semiseparable matrices without knowing their inverses to be tridiagonal. During this historical evolution the definition of semiseparable matrices has always been a difficult point leading to misunderstandings; sometimes they were defined as the inverses of irreducible tridiagonal matrices leading to generator representable matrices, while in other cases they were defined as matrices having low rank blocks below the diagonal.

show abstract

“…There are also algorithms that solve Toeplitz systems of linear equations with the complexity O(n log 2 n) (see, e.g., [5,8] and the lists of references given there), but in the context of our problem their significance is only theoretical. Moreover, since G n is associated with the forward difference operator (see (2.4) and (3.7)), in some cases it can be inverted using explicit formulas (see, e.g., [2,18]). Now, let us assume that k, l ≪ n which is the most common case for our problem.…”

Section: An Iterative Approximate Methods Of Solving Boundary Value P...mentioning

confidence: 99%

An iterative approximate method of solving boundary value problems using dual Bernstein polynomials

Gospodarczyk,

Woźny

2017

Preprint

View full text Add to dashboard Cite

In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties of dual Bernstein polynomials which guarantee high efficiency of our approach. The method can deal with both linear and nonlinear differential equations. Moreover, not only second order differential equations can be solved but also higher order differential equations. Illustrative examples confirm the versatility of our method.

show abstract

Exact inverses of certain band matrices

Abstract: Summary.A method for determining exact inverses for arbitrary size band matrices of Toeplitz type and closely related types is outlined. A number of examples arising in statistical problems and finite differences are considered and the initial required elements of the inverses are given.

Cited by 21 publications

References 3 publications

Real-time single-molecule 3D tracking in E. coli based on cross-entropy minimization

Real-time single-molecule 3D tracking in E. coli based on cross-entropy minimization

A bibliography on semiseparable matrices*

An iterative approximate method of solving boundary value problems using dual Bernstein polynomials

Contact Info

Product

Resources

About